Conservation laws continuity equation

The phenomenology of model B, where (j) is conserved, can also be outlined simply. Since (j) is conserved, it obeys a conservation law (continuity equation) ... 

According to the law of mass conservation, the continuity equation of the water are derived as given in eq. (2). For simplify, the equation ie expressed along the main seepage directions denoted as coordinate axes x, y, z ... 

The tliree conservation laws of mass, momentum and energy play a central role in the hydrodynamic description. For a one-component system, these are the only hydrodynamic variables. The mass density has an interesting feature in the associated continuity equation the mass current (flux) is the momentum density and thus itself is conserved, in the absence of external forces. The mass density p(r,0 satisfies a continuity equation which can be expressed in the fonn (see, for example, the book on fluid mechanics by Landau and Lifshitz, cited in the Furtlier Reading)... 

The continuity equation is the expression of the law of conservation of mass. This equation is written as... 

The continuity equation is a mathematical formulation of the law of conservation of mass of a gas that is a continuum. The law of conservation of mass states that the mass of a volume moving with the fluid remains unchanged... 

From a physical point of view, the finite difference method is mostly based based on the further replacement of a continuous medium by its discrete model. Adopting those ideas, it is natural to require that the principal characteristics of a physical process should be in full force. Such characteristics are certainly conservation laws. Difference schemes, which express various conservation laws on grids, are said to be conservative or divergent. For conservative schemes the relevant conservative laws in the entire grid domain (integral conservative laws) do follow as an algebraic corollary to difference equations. 

Let us consider a dynamically symmetric binary mixture described by the scalar order parameter field < )(r) that gives the local volume fraction of component A at point r. The order parameter < )(r) should satisfy the local conservation law, which can be written as a continuity equation [143] ... 

Finally, continuity equations that account for mass conservation laws. 

In order to calculate the density of reactant B about A, it is necessary to know by what means the reactants migrate in solution. Under most circumstances, diffusion is a very adequate description (the limitations of and complications to diffusion are discussed in Sect. 6, Chap. 8 Sect. 2 and Chap. 11). In this simple analysis of diffusion, Fick s laws will be used with little further justification, save to note that Fick s second law is identical to the equation satisfied by a random walk function. Hardly a surprising result, because diffusion is a random walk with no retention of information about where the diffusing species was before its current location. In Chap. 3 Sect. 1, the diffusion equation is derived from thermodynamic considerations and the continuity equation (law of conservation of mass). 

Beginning with a mass-conservation law, the Reynolds transport theorem, and a differential control volume (Fig. 4.30), derive a steady-state mass-continuity equation for the mean circumferential velocity W in the annular shroud. Remember that the pressure p 6) (and hence the density p(6) and velocity V(6)) are functions of 6 in the annulus. 

Deriving the mass-continuity equation begins with a mass-conservation principle and the Reynolds transport theorem. Unlike the channel with chemically inert walls, when surface chemistry is included the mass-conservation law for the system may have a source term,... 

Turn now to the individual species continuity equations where the mass-conservation law for the system includes both homogeneous- and heterogeneous-chemistry source terms,... 

Deriving the governing equations begins with the underlying conservation laws and the Reynolds transport theorem. Consider first the overall mass continuity, where... 

Consider next the individual-species continuity equation, where the extensive variable is the mass of species k, and the associated intensive variable is the mass fraction Yk. The conservation law follows as... 

Conservation Laws. The fundamental conservation laws of physics can be used to obtain the basic equations of fluid motion, the equations of continuity (mass conservation), of flow (momentum conservation), of... 

Differential equations or a set of differential equations describe a system and its evolution. Group symmetry principles summarize both invariances and the laws of nature independent of a system s specific dynamics. It is necessary that the symmetry transformations be continuous or specified by a set of parameters which can be varied continuously. The symmetry of continuous transformations leads to conservation laws. 

This model is applicable if the thickness of the boundary layer x 2(Dt)1/2 is small as compared to the distance between the reactor walls (where t is the residence time in the reactor and D the diffusion coefficient). In this case, component concentration equations are obtained from the mass and momentum conservation laws and the continuity equation... 

The law of conservation of mass can be written in the form of the continuity equation... 

The region over which this balance is invoked is the heterogeneous porous catalyst pellet which, for the sake of simplicity, is described as a pscudohomoge-ncous substitute system with regular pore structure. This virtual replacement of the heterogeneous catalyst pellet by a fictitious continuous phase allows a convenient representation of the mass and enthalpy conservation laws in the form of differential equations. Moreover, the three-dimensional shape of the catalyst pellet is replaced by assuming a one-dimensional model... 

The principles and basic equations of continuous models have already been introduced in Section 6.2.2. These are based on the well known conservation laws for mass and energy. The diffusion inside the pores is usually described in these models by the Fickian laws or by the theory of multicomponent diffusion (Stefan-Maxwell). However, these approaches basically apply to the mass transport inside the macropores, where the necessary assumption of a continuous fluid phase essentially holds. In contrast, in the microporous case, where the pore size is close to the range of molecular dimensions, only a few molecules will be present within the cross-section of a pore, a fact which poses some doubt on whether the assumption of a continuous phase will be valid. 

Clausius/Clapeyron equation, 182 Coefficient of performance, 275-279, 282-283 Combustion, standard heat of, 123 Compressibility, isothermal, 58-59, 171-172 Compressibility factor, 62-63, 176 generalized correlations for, 85-96 for mixtures, 471-472, 476-477 Compression, in flow processes, 234-241 Conservation of energy, 12-17, 212-217 (See also First law of thermodynamics) Consistency, of VLE data, 355-357 Continuity equation, 211 Control volume, 210-211, 548-550 Conversion factors, table of, 570 Corresponding states correlations, 87-92, 189-199, 334-343 theorem of, 86... 

This equation can be applied to the total mass involved in a process or to a particular species, on either a mole or mass basis. The conservation law for mass can be applied to steady-state or unsteady-state processes and to batch or continuous systems. A steady-state system is one in which there is no change in conditions (e.g., temperature, pressure) or rates of flow with time at any given point in the system the accumulation term then becomes zero. If there is no chemical reaction, the generation term is zero. All other processes are classified as unsteady state. 

For each continuous phase k present in a multiphase system consisting of N phases, in principle the set of conservation equations formulated in the previous section can be applied. If one or more of the N phases consists of solid particles, the Newtonian conservation laws for linear and angular momentum should be used instead. The resulting formulation of a multiphase system will be termed the local instant formulation. Through the specification of the proper initial and boundary conditions and appropriate constitutive laws for the viscous stress tensor, the hydrodynamics of a multiphase system can in principle be obtained from the solution of the governing equations. 

Continuing with this analysis, suppose that the equations of motion given in (58) possess a set of n-c conservation laws, Ak(x), k = l,...,nc, which satisfy the relation ... 

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